CHAPTER AT A GLANCE

Contents:

(i)                                        Definition of rest and motion

(ii)                                      Types of motion

(iii)                                    Types of physical quantities

(iv)                                   Distance, displacement and their differences

(v)                                     Uniform and non-uniform motion and their types

(vi)                                   Speed and velocity

(vii)                                 Acceleration, decelerated motion

(viii)                               Graphical plotting of uniform and non-uniform motion

(ix)                                   Equation of motion and their derivation

Rest: A body is said to be in a state of rest when its position does not change with respect to a reference point.

Motion: A body is said to be in a state of motion when its position change continuously with reference to a point.

Motion can be of different types depending upon the type of path by which the object is going through.

(i)          Circulatory motion/Circular motion – In a circular path.

(ii)        Linear motion – In a straight line path.

(iii)      Oscillatory/Vibratory motion – To and fro path with respect to origin.

Scalar quantity: It is the physical quantity having own magnitude but no direction e.g., distance, speed.

Vector quantity: It is the physical quantity that requires both magnitude and direction e.g., displacement, velocity.

Distance and Displacement:

•              The actual path or length travelled by an object during its journey from

its initial position to its final position is called the distance.

•              Distance is a scalar quantity which requires only magnitude but no direction to explain it.It can only be positive.

Example, Ramesh traveled 65 km (Distance is measured by odometer in vehicles.)

•              Displacement is a vector quantity requiring both magnitude and direction for its explanation.It can be negative, positive or even zero.

Example, Ramesh travelled 65 km south-west from Clock Tower.

•              Displacement can be zero (when initial point and final point of motion

Are same) Example, circular motion.

Uniform and Non-uniform Motion

•          Uniform Motion :

When a body travels equal distance in equal interval of time, then the motion is said to be uniform motion.

•         Non-uniform Motion:

In this type of motion, the body will travel unequal distances in equal intervals of time.

Non-uniform motion is of two types:

     

(i) Accelerated Motion: When motion of a body increases with time.

        

(i)   De-accelerated Motion: When motion of a body decreases with time.                

Speed

• Speed is defined as the total distance travelled by the object in the time interval during which the motion takes place. SI unit of speed is meter per second. So,
  
  where s is the distance travelled by the body and t is the time taken by the body to travel distance s.
• Speed of a body gives us the idea how slow or fast that body is moving.
• The ratio of total distance to total time taken by the body gives its average speed. The speed of a body at a given instant is its instantaneous speed.
  
• A body is said to have constant or uniform speed if it travels equal distance in equal intervals of time.

Velocity

• The rate of change of displacement of a body with the passage of time is known as velocity of the body. Velocity of an object is measured in meter per second in SI units. So,
  
• Velocity is nothing but the speed of an object moving in a definite direction.
• The velocity of an object can be uniform or variable. It can be changed by changing the objects speed, direction of motion or both.
• So velocity of a body is a vector quantity involving both distance and displacement whereas speed of a body is a scalar quantity and it only has magnitude and does not have specific direction.
• Thus a body is said to be moving with uniform velocity if it covers equal distances in equal intervals of time in a specified direction.
• A body is said to be moving with non uniform velocity if it covers unequal distances in equal intervals of time and vice-versa in a specified direction or if it changes the direction of motion.
• The velocity of a body can be changed in two ways first by changing the speed of the body and second by changing the direction of motion of the body by keeping the speed constant. Also both speed and direction of the body can be varied in order to change the velocity of the body.
• When velocity of the object changes at a uniform rate, then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time. That is,
  
  Where u is the initial velocity of the object and v is the final velocity of the object.

Acceleration

• Acceleration is a measure of the change in the velocity of an object per unit time and mathematically it is given as
  
• If the velocity of an object changes from an initial value u to the final value v in time t, the acceleration a is given by,
  a=v−ut
  and this kind of motion is called accelerated motion.
• A body has uniform acceleration if it travels in a straight line and its velocity increases by equal amount in equal intervals of time for example freely falling bodies, motion of ball rolling down the inclined plane etc.
• A body has non uniform acceleration if its velocity increases or decreases by unequal amount in equal intervals of time.
• If acceleration is in the direction of the velocity then it is positive acceleration and if it is in the direction opposite to the direction of velocity then it is negative and the negative acceleration is termed retardation.
• SI unit of acceleration is ms^-2.

Equations of uniformly accelerated motion

There are three equation of motion : v = u + at, s = ut + 1/2at2 and v2 = u2 + 2as can be derived with help of the graph.

Let an object is moving with uniform acceleration.

Let the initial velocity of the object = u

Let the object is moving with uniform acceleration, a.

Let object reaches at point B after time, t and its final velocity becomes, v

Draw a line parallel to x-axis DA from point, D from where object starts moving.

Draw another line BA from point B parallel to y-axis which meets at E at y-axis.

Let OE = time, t

Now, from the graph,

BE = AB + AE

⇒ v = DC + OD (Since, AB = DC and AE = OD)

⇒ v = DC + u (Since, OD = u)

⇒ v = DC + u ------------------- (i)

Now,Acceleration(a)=ChangeinvelocityTimetaken

⇒a=v−ut=OC−ODt=DCt

⇒at=DC.............(ii)

By substituting the value of DC from (ii) in (i) we get 

v = at +u

⇒ v = u + at

Above equation is the relation among initial velocity (u), final velocity (u), acceleration (a) and time (t). It is called first equation of motion.

 2nd equation of Motion :

Equation for distance –time relation:

Distance covered by the object in the given time ‘t’ is given by the area of the trapezium ABDOE

Let in the given time, t the distance covered by the moving object = s

The area of trapezium, ABDOE

= Distance (s) = Area of ΔABD + Area of ADOE

⇒12×AB×AD+(OD×OE)

⇒s=12×DC×AD+(u+t)[Since,AB=DC]

⇒s=12×at×t+ut[Since,DC=atfromequation(ii)]

⇒s=12at2+ut

⇒s=ut+12at2

The above expression gives the distance covered by the object moving with uniform acceleration. This expression is known as second equation of motion.

Equation for Distance Velocity Relation: Third equation of Motion:

The distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDO

Therefore,

Area of trapezium ABDOE

=12 sum of parallel sides + distance between parallel sides 

⇒Distance (s) =12(DO+BE)×OE

⇒s=12(u+v)×t..............(iii)

Now,fromequation(ii)a=v−ut

Therefore,t=v−ua.......(iv)

After substituting the value of t from equation (iv) in equation (iii) 

⇒s=12(u+v)×(v−u)a

⇒s=12a(v+u)(v−u)

⇒2as=(v+u)(v−u)

⇒2as+u2=v2

⇒v2=u2+2as

The above expression gives the relation between position and velocity and is called the third equation of motion.

Uniform circular motion

• When an object moves in a circular path at a constant speed then motion of the object is called uniform circular motion.
• In our everyday life ,we came across many examples of circular motion for example cars going round the circular track and many more .Also earth and other planets revolve around the sun in a roughly circular orbits
• If the speed of motion is constant for a particle moving in a circular motion still the particles accelerates because of constantly changing direction of the velocity.
• If an object moves in a circular path with uniform speed, its motion is called uniform circular motion
• Here in circular motion, we use angular velocity in place of velocity we used while studying linear motion.
• Force which is needed to make body travel in a circular path is called centripetal force.
• We know that the circumference of a circle of radius r is given by 2πr. If the body takes t seconds to go once around the circular path of radius r, the velocity v is given by
  v=2πrt
• One thing we must keep in mind is that uniform linear motion is not accelerated but uniform circular motion is accelerated motion.


• Examples of uniform circular motion are 
  (a) Motion of artificial satellites around the earth
  (b) Moon, the natural satellite of earth, moves in uniform circular motion round the earth.
  (c) Cyclist moving on a circular track with a constant speed exhibits uniform circular motion.